Let A be a finite abelian group.
For any prime number p, let Ap be the set of
elements g of A
such that png = 0 for some natural number p
(thus Ap is the set of all "p-elements" of A).
- Ap is a subgroup of A.
- A is the direct sum of these subgroups Ap.
-
If A and A' are finite abelian groups and f:A → A' is
a group homomorphism,
then f(Ap) is contained in
A'p.
-
In particular: If B is a subgroup of A, then
Bp is contained in
Ap.
Thus, dealing with subgroups of finite abelian groups,
we may restrict to consider p-groups.